cp's OEIS Frontend

A362369 Triangle read by rows, T(n, k) = binomial(n, k) * k! * Stirling2(n-k, k), for n >= 0 and 0 <= k <= n//2, where '//' denotes integer division.

%I A362369 #11 May 10 2023 08:24:01
%S A362369 1,0,0,2,0,3,0,4,12,0,5,60,0,6,210,120,0,7,630,1260,0,8,1736,8400,
%T A362369 1680,0,9,4536,45360,30240,0,10,11430,216720,327600,30240,0,11,28050,
%U A362369 956340,2772000,831600,0,12,67452,3993000,20207880,13305600,665280
%N A362369 Triangle read by rows, T(n, k) = binomial(n, k) * k! * Stirling2(n-k, k), for n >= 0 and 0 <= k <= n//2, where '//' denotes integer division.
%F A362369 From _Mélika Tebni_, May 10 2023: (Start)
%F A362369 E.g.f. of column k: (x*(exp(x)-1))^k / k!.
%F A362369 Sum_{k=0..n-1} (-1)^(n+k-1)*T(n+k-1, k) = A000169(n), for n > 0. (End)
%e A362369 Triangle T(n, k) starts:
%e A362369 [0] 1;
%e A362369 [1] 0;
%e A362369 [2] 0, 2;
%e A362369 [3] 0, 3;
%e A362369 [4] 0, 4,   12;
%e A362369 [5] 0, 5,   60;
%e A362369 [6] 0, 6,  210,   120;
%e A362369 [7] 0, 7,  630,  1260;
%e A362369 [8] 0, 8, 1736,  8400,  1680;
%e A362369 [9] 0, 9, 4536, 45360, 30240;
%p A362369 T := (n, k) -> binomial(n, k) * k! * Stirling2(n-k, k):
%p A362369 seq(seq(T(n, k), k = 0..iquo(n, 2)), n = 0..9);
%p A362369 # second program:
%p A362369 egf := k-> (x*(exp(x)-1))^k / k!:
%p A362369 A362369 := (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
%p A362369 seq(print(seq(A362369(n, k), k=0..iquo(n,2))), n=0..12); # _Mélika Tebni_, May 10 2023
%o A362369 (SageMath)
%o A362369 def A362369(n, k):
%o A362369     return binomial(n, k) * factorial(k) * stirling_number2(n - k, k)
%o A362369 for n in range(10):
%o A362369     print([A362369(n, k) for k in range(n//2 + 1)])
%Y A362369 Cf. A000169, A052506 (row sums), A362788, A362789.
%K A362369 nonn,tabf
%O A362369 0,4
%A A362369 _Peter Luschny_, May 04 2023